where \((T,\lambda )\)  are tuning parameters and for some \(\eta \ge 0\)  we define \(\mathcal {B}(\eta ):=\lbrace \textbf {h}\in \mathbb {H}, \Vert \textbf {h}-\textbf {g}_1\Vert \le \eta \rbrace \)  to be a small ball containing \(\textbf {g}_1\) . The constraint \(\Vert \textbf {h}\Vert _1\le T\)  might seem unnecessary or at least redundant with the penalty \(\lambda \Vert \textbf {h}\Vert _1\) , we impose it due to the non-convexity of \(\widehat{R}_\phi \) . From a theoretical point of view, it is essential to derive guarantees on \(\widehat{\textbf {g}}\) . In the sequel we will show that \(T=O((\frac{n}{\log (pD)})^{1/4})\)  is convenient. Thus, this constraint is not restrictive. Note that similar constraints were found in [1]} and [2]}.
